Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (openai/gpt-5.4)audited 2026-07-25
How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced — the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted — a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated — a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Left successor law for addition

Statement

For all m,n∈Nm, n \in \mathbb{N}: σ(m)+n=σ(m+n)\sigma(m) + n = \sigma(m + n). (The recursion defines addition on the second argument; this is the matching law for adding on the first.)

Facts & Assumptions

Given: Addition with m+0=mm + 0 = m and m+σ(k)=σ(m+k)m + \sigma(k) = \sigma(m + k) (Addition of natural numbers).

[L1]

Induction principle (The principle of mathematical induction).

Proof

technique · induction
1.1

Base n=0n = 0: σ(m)+0=σ(m)\sigma(m) + 0 = \sigma(m) by x+0=xx + 0 = x, and σ(m+0)=σ(m)\sigma(m + 0) = \sigma(m) by m+0=mm + 0 = m, so σ(m)+0=σ(m+0)\sigma(m) + 0 = \sigma(m + 0).

base
1.2

Inductive hypothesis: σ(m)+n=σ(m+n)\sigma(m) + n = \sigma(m + n).

ih
2.1

Step: σ(m)+σ(n)=σ(σ(m)+n)=σ(σ(m+n))=σ(m+σ(n))\sigma(m) + \sigma(n) = \sigma(\sigma(m) + n) = \sigma(\sigma(m + n)) = \sigma(m + \sigma(n)), applying x+σ(k)=σ(x+k)x + \sigma(k) = \sigma(x + k) at the outer sum, then the hypothesis, then m+σ(n)=σ(m+n)m + \sigma(n) = \sigma(m + n) backward inside σ\sigma.

step 1.2given
3.1

By induction [L1], σ(m)+n=σ(m+n)\sigma(m) + n = \sigma(m + n) for all nn, and since mm was arbitrary, for all m,n∈Nm, n \in \mathbb{N}.

step 1.1step 2.1discharge-induction∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 15 results over 9 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources